Topological quantum numbers and curvature -- examples and applications

Using the idea of the degree of a smooth mapping between two manifolds of the same dimension we present here the topological (homotopical) classification of the mappings between spheres of the same dimension, vector fields, monopole and instanton solutions. Starting with a review of the elements of Riemannian geometry we also present an original elementary proof of the Gauss-Bonnet theorem and the Poincar\'{e}-Hopf theorem.